Autr Kw Bejmi Rigsby http://m.mthforcollege.com Trsformig Numericl Methods Eductio for STEM Udergrdutes
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. solve set of simulteous lier equtios usig Nïve Guss elimitio,. ler the pitflls of the Nïve Guss elimitio method,. uderstd the effect of roud-off error whe solvig set of lier equtios with the Nïve Guss elimitio method,
4. ler how to modify the Nïve Guss elimitio method to the Gussi elimitio with prtil pivotig method to void pitflls of the former method,. fid the determit of squre mtri usig Gussi elimitio, d 6. uderstd the reltioship betwee the determit of coefficiet mtri d the solutio of simulteous lier equtios.
A method to solve simulteous lier equtios of the form [A][X][C] Two steps. Forwrd Elimitio. Bck Substitutio
.7 96. 6.8.7.6 4.8 79. 77. 6.8 44 8 64 The gol of forwrd elimitio is to trsform the coefficiet mtri ito upper trigulr mtri
A set of equtios d ukows + + +... + b + + +... + b + + +... +...... b (-) steps of forwrd elimitio
Step For Equtio, divide Equtio by by. d multiply ( + + +... + b ) + +... + b
Subtrct the result from Equtio. + + +... + b + +... + b... b b + + or +... + b
Repet this procedure for the remiig equtios to reduce the set of equtios s + + +... + b + +... + b + +... + b......... + + +... Ed of Step b
Step Repet the sme procedure for the rd term of Equtio. + + +... + b + +... + b + " " "... + b...... + + b " " "... Ed of Step
At the ed of (-) Forwrd Elimitio steps, the system of equtios will look like + + +... + b + +... + b " " " +... + b ( ) ( ) b...... Ed of Step (-)
) ( " ) ( " " b b b b
.7 96. 6.8.7.6 4.8 Solve ech equtio strtig from the lst equtio Emple of system of equtios
+ + +... + b + +... + b " " +... + b...... " ( ) ( ) b
Strt with the lst equtio becuse it hs oly oe ukow b ( ) ( )
( ) ( ) ( ) ( ) ( ),..., for...,,, + + + + i b i ii i i i i i i i i i i i i i ( ) ( ) ( ),..., for + i b i ii i j j i ij i i i ) ( ) ( b
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The upwrd velocity of rocket is give t three differet times Tble Velocity vs. time dt. Time, t ( s) Velocity, v ( m/s) 6.8 8 77. 79. The velocity dt is pproimted by polyomil s: v ( ) t t + t +, t. Fid the velocity t t6 secods.
( ). t, t t t v + + Assume v v v t t t t t t Results i mtri templte of the form: Usig dt from Tble, the mtri becomes: 79. 77. 6.8 44 8 64
79 44 77 8 64 6 8 79 77 6 8 44 8 64....... Forwrd Elimitio. Bck Substitutio
79 44 77 8 64 6 8 79 77 6 8 44 8 64....... Forwrd Elimitio. Bck Substitutio
Forwrd Elimitio
Number of steps of forwrd elimitio is (-)(-)
64 44 8. 6.8 77. 79. Divide Equtio by d 64 multiply it by 64,.6. [ 6.8].6 [ 64.8.6 7.48] Subtrct the result from Equtio [ 64 8 77.] [ 64.8.6 7.48] [ 4.8.6 96.8] Substitute ew equtio for Equtio 44 4.8.6 6.8 96.8 79.
44 4.8.6. 6.8 96.8 79. [ 6.8].76 [ 44 8.8.76 6.68] Subtrct the result from Equtio Substitute ew equtio for Equtio [ 44 79.] [ 44 8.8.76 6.68] [ 6.8 4.76.968] Divide Equtio by d 44 multiply it by 44,.76. 4.8 6.8.6 4.76 6.8 96.8.968
4.8 6.8.6 4.76 6.8 96.8.968 Divide Equtio by 4.8 d multiply it by 6.8, 6.8.. 4.8 [ 4.8.6 96.8]. [ 6.8.46 6.78] [ 6.8 4.76.968] Subtrct the result from Equtio [ 6.8.46 6.78] [.7.76] Substitute ew equtio for Equtio 4.8.6.7 6.8 96.8.76
Bck Substitutio
4.8.6.7 6.8 96..7 4.8.6.7 6.8 96.8.76 Solvig for.7.76.76.7.87
Solvig for 9.69 4.8.87.6 96.8 4.8.6 96.8 96.8.6 4.8 + +.76 96.8 6.8.7.6 4.8
4.8.6.7 6.8 96..76 Solvig for + + 6.8 6.8 6.8 9.69.87.947
79 77 6 8 44 8 64....87 9.69.947
Solutio The solutio vector is.947 9.69.87 The polyomil tht psses through the three dt poits is the: v ( t) t + t +.947t + 9.69t +.87, t v ( ) ( ) 6.947 6 + 9.69( 6) 9.686 m/s. +.87
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9 6 7 + + + 9 6 7 Pitfll#. Divisio by zero
9 4 6 7 + + + + 9 4 6 7
8 4 4 6 7 + + + + 8 4 4 6 7 6. 9 6. 7 Divisio by zero is possibility t y step of forwrd elimitio
Pitfll#. Lrge Roud-off Errors 9.7 4 7.49 Ect Solutio
Pitfll#. Lrge Roud-off Errors.99999..96 9.7 4 7.49 Solve it o computer usig 6 sigifict digits with choppig
Pitfll#. Lrge Roud-off Errors.9999..6 9.7 4 7.49 Solve it o computer usig sigifict digits with choppig Is there wy to reduce the roud off error?
Icrese the umber of sigifict digits Decreses roud-off error Does ot void divisio by zero
Gussi Elimitio with Prtil Pivotig Avoids divisio by zero Reduces roud off error
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Possible divisio by zero Lrge roud-off errors
Icrese the umber of sigifict digits Decreses roud-off error Does ot void divisio by zero
Gussi Elimitio with Prtil Pivotig Avoids divisio by zero Reduces roud off error
At the begiig of the k th step of forwrd elimitio, fid the mimum of,..., kk, k +, k k If the mimum of the vlues is pk i the p th row, k p, the switch rows p d k.
4 b b b b
9 8 6 4 7 8 6 9 4 6 7 6.7. 4 6 4 Which two rows would you switch?
6 9 8 6 7 8 6 9 4 4 7 6.7. 4 6 4 Switched Rows
A method to solve simulteous lier equtios of the form [A][X][C] Two steps. Forwrd Elimitio. Bck Substitutio
Sme s ïve Guss elimitio method ecept tht we switch rows before ech of the (-) steps of forwrd elimitio.
4 b b b b
) ( " ) ( " " b b b b
+... + b + + +... + b + " " " +... + b ( ) ( ) b......
b ( ) ( ) i b ( i) ( i) i j ii ij j i+ ( ) for i,..., i
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79 77 6 8 44 8 64... Solve the followig set of equtios by Gussi elimitio with prtil pivotig
79 77 6 8 44 8 64.... Forwrd Elimitio. Bck Substitutio 79. 44 77. 8 64 6.8
Forwrd Elimitio
Number of steps of forwrd elimitio is (-)(-)
Emie bsolute vlues of first colum, first row d below., 64, 44 Lrgest bsolute vlue is 44 d eists i row. Switch row d row. 64 44 8 6.8 77. 79. 44 64 8 79. 77. 6.8
44 64 8 79. 77. 6.8 [ 44 79.].4444 [ 6.99..4444 4.] Subtrct the result from Equtio Divide Equtio by 44 d 64 44 multiply it by 64,.4444. [ 64 8 77.] [ 6.99..4444 4.] [.667.6.] Substitute ew equtio for Equtio 44.667.6 79.. 6.8
44.667.6 79.. 6.8 [ 44 79.].76 [..8.76 48.47] Subtrct the result from Equtio Substitute ew equtio for Equtio Divide Equtio by 44 d multiply it by,.76. [ 6.8] [.8.76 48.47] [.97.864 8.] 44.667.97.6.864 44 79.. 8.
Emie bsolute vlues of secod colum, secod row d below..667,.97 Lrgest bsolute vlue is 44 d eists i row. Switch row d row. 44.667.97.6.864 79.. 8. 44.97.667.864.6 79. 8..
44.97.667.864.6 79. 8.. [.97.864 8.].94 [.667.76.] Subtrct the result from Equtio Substitute ew equtio for Equtio Divide Equtio by.97 d multiply it by.667,.94. [.667.6.] [.667.76.] [..] 44.97.667.97 79..864 8...
Bck Substitutio
..... Solvig for 8 79 864 97 44.. 8..864.97 79. 44......
44. 97 79.. 864 8... Solvig for.97 +.864 8. 8..864.97 8..864..97 9.67
Solvig for.97 44. 9.67 79. 44 79. 79. 44 + + 8 79 864 97 44......
79 77 6 8 44 8 64.... 9.67.97
6.9 6.99 7 7 + + + Cosider the system of equtios I mtri form 6.99 7 6.9 7 Solve usig Gussi Elimitio with Prtil Pivotig usig five sigifict digits with choppig
Forwrd Elimitio: Step Emiig the vlues of the first colum, -, d or,, d The lrgest bsolute vlue is, which mes, to follow the rules of Prtil Pivotig, we switch row with row. 6.9 7 6.99 7. 6. 7. 6. 7 Performig Forwrd Elimitio
6.. 7 6.. 7 Forwrd Elimitio: Step Emiig the vlues of the first colum -. d. or. d. The lrgest bsolute vlue is., so row is switched with row. 6. 7. 6. 7 Performig the row swp
Forwrd Elimitio: Step Performig the Forwrd Elimitio results i: 6.. 7 6.. 7
Bck Substitutio Solvig the equtios through bck substitutio 7. 6. 7. 6. 7 + 7 6. 6...
[ ] X ect [ ] X clculted Compre the clculted d ect solutio. The fct tht they re equl is coicidece, but it does illustrte the dvtge of Prtil Pivotig
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Determit of Squre Mtri Usig Nïve Guss Elimitio Emple
If multiple of oe row of [A] is dded or subtrcted to other row of [A] to result i [B] the det(a)det(b)
The determit of upper trigulr mtri [A] det A... ii... give by ( ) i is ii
Usig forwrd elimitio to trsform [A] upper trigulr mtri, [U]. to [ A ] [ ] U ( A) det( U ) det
Usig ïve Gussi elimitio fid the determit of the followig squre mtri. 64 44 8
Forwrd Elimitio
64 44 8.6 [ ] [ 64.8.6] Subtrct the result from Equtio Substitute ew equtio for Equtio. Divide Equtio by d [ 64 8 ] [ 64.8.6] [ 4.8.6] 44 64 multiply it by 64,. 4.8.6.6
44 4.8..6 [ ].76 [ 44 8.8.76] Subtrct the result from Equtio Substitute ew equtio for Equtio Divide Equtio by d multiply it by 44,.76. [ 44 ] [ 44 8.8.76] [ 6.8 4.76] 4.8 6.8 44.6 4.76
4.8 6.8..6 4.76 ([ 4.8.6] ). [ 6.8.46] Subtrct the result from Equtio Substitute ew equtio for Equtio Divide Equtio by 4.8 d multiply it by 6.8, 6.8.. 4.8 [ 6.8 4.76] [ 6.8.46] [.7] 4.8.6.7
After forwrd elimitio 64 44 8 4.8.6.7 ( A) det u u u 84. ( 4.8).7
Forwrd Elimitio Bck Substitutio Pitflls Improvemets Prtil Pivotig Determit of Mtri
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